By Bjorn Poonen, Yuri Tschinkel

One of many nice successes of 20th century arithmetic has been the striking qualitative figuring out of rational and fundamental issues on curves, gleaned partially in the course of the theorems of Mordell, Weil, Siegel, and Faltings. It has turn into transparent that the examine of rational and vital issues has deep connections to different branches of arithmetic: complicated algebraic geometry, Galois and ,tale cohomology, transcendence concept and diophantine approximation, harmonic research, automorphic varieties, and analytic quantity idea. this article, which makes a speciality of better dimensional kinds, presents accurately such an interdisciplinary view of the topic. it's a digest of analysis and survey papers by means of top experts; the booklet files present wisdom in higher-dimesional mathematics and offers symptoms for destiny examine. will probably be beneficial not to purely to practitioners within the box, yet to a large viewers of mathematicians and graduate scholars with an curiosity in mathematics geometry. individuals comprise: P. Swinnerton-Dyer * B. Hassett * Yu. Tschinkel * J. Shalika * R. Takloo-Bighash * J.-L. Colliot-Th,lSne * A. de Jong * Ph. Gille * D. Harari * J. Harris * B. Mazur * W. Raskind * J. Starr * T. Wooley

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Soc. (to appear). [9] Hilbert’s Tenth Problem: Relations with Arithmetic and Algebraic Geometry (ed. Jan Denef et al), Contemporary Mathematics, vol. 270. [10] Mathematical Developments arising from Hilbert Problems, AMS Symposia in Pure Mathematics, Vol XXVIII (ed. Browder), (Providence, 1976). Cassels, Second descents for elliptic curves, J. reine angew. Math. 494(1998), 101-127. Sansuc and Sir Peter Swinnerton-Dyer, Intersections of two quadrics and Chˆatelet surfaces, J. reine angew. Math. 373(1987), 37-107 and 374(1987), 72-168.

357-404 (Birkh¨auser, 2001). [44] Sir Peter Swinnerton-Dyer, The solubility of diagonal cubic surfaces, ´ Norm. Sup. (4) 34(2001), 891-912. Ann. Scient. Ec. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, S´em. Bourbaki 306(1966). , Cambridge, 1997).

Thus finiteness implies that if X contains at most p − 1 elements of order exactly p for some prime p then it actually contains no such elements; hence an element which is killed by p is trivial, and the curves of genus 1 in that equivalence class contain points defined over K. For use later, we state the case p = 2 as a lemma. Lemma 3 Suppose that X(J) is finite and the quotient of the 2-Selmer group of J by its soluble elements has order at most 2; then that quotient is actually trivial. 17 This result will play a crucial role in §§6 and 7.